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Chaos and Fractals: New Frontiers of Science

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The first edition of this vast introduction to chaos and fractals appeared in 1992. This new edition is virtually identical to the original except for some material that has been removed. The first edition contained two appendices, one on fractal image compression by Yuval Fisher, and another on multifractal measures by Carl J. G. Evertsz and Benoit B. Mandelbrot. (The information in the first appendix is now contained in a book by Fisher.) Also, each chapter of the first edition contained a "Program of the Chapter,: written in BASIC; these have been replaced by applets on the authors' website at

The biggest change I could find consisted of two new sentences at the end of the book. Also, a table on the approximation of π by computer has been updated. The number of references has gone down from 320 to 281, but I saw no new references, leading the reader to wonder whether anything new has been done in chaos and fractals since 1992. The preface mentions that several errors in the first edition have been corrected.

Despite the lack of new material, the book is still a wonderful tour of a fascinating area of mathematics, and now the reader can take this tour while carrying around a slimmer (but still hefty) volume. The authors aim for a general audience, although they assume a knowledge of such topics as the algebra of functions and logarithms, sums of infinite geometric series, and limits. In the section on continuous dynamical systems, they define derivative and differential equation, and then go on to use those concepts. Similarly, the chapter on Julia sets offers a short introduction to complex numbers. More advanced mathematical ideas are contained in side comments that sometimes go on for more than a page. Discerning where a side comment ends and the main text begins can be confusing unless the reader is alert to the sans serif font used in the side comments. Technical terms (such as compact) are defined in footnotes.

The fourteen chapters cover almost every aspect of chaos and fractals, including self-similarity, fractal dimension, transformations, the chaos game, recursive structures, cellular automata, Brownian motion, period doubling, strange attractors, Julia sets, and the Mandelbrot set. There is a bit of symbolic dynamics, although that term is not introduced.

There are hundreds of illustrations, including 40 gorgeous color plates. The book is full of quotes and historical tidbits that make for entertaining reading. Much of the material is elementary enough for a novice to grasp, yet the coverage is sufficiently broad and thorough that almost every mathematician will learn something new. The authors have a friendly, conversational style, often asking questions of the reader to indicate the significance of what is to come next.

This is a great book, despite the lack of new material in the new edition.

Raymond N. Greenwell ( is a Professor of Mathematics at Hofstra University in Hempstead, New York. His research interests include applied mathematics and statistics, and he is coauthor of the texts Finite Mathematics and Calculus with Applications, published by Addison Wesley.


Date Received: 
Wednesday, December 1, 2004
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Heinz-Otto Peitgen, Hartmut Jürgens, and Dietmar Saupe
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Raymond N. Greenwell
Introduction: Causality Principle, Deterministic Laws and Chaos.- The Backbone of Fractals: Feedback and the Iterator.- Classical Fractals and Self-Similarity.- Limits and Self-Similarity.- Length, Area, and Dimension: Measuring Complexity and Scaling Properties.- Encoding Images by Simple Transformations.- The Chaos Game: How Randomness Creates Deterministic Shapes.- Recursive Structures: Growing Fractals and Plants.- Pascal's Triangle: Cellular Automata and Attractors.- Irregular Shapes: Randomness in Fractal Constructions.- Deterministic Chaos: Sensitivity, Mixing, and Periodic Points.- Order and Chaos: Period-Doubling and Its Chaotic Mirror.- Strange Attractors: The Locus of Chaos.- Julia Sets: Fractal Basin Boundaries.- The Mandelbrot Set: Ordering the Julia Set.
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Wednesday, May 24, 2006